Wednesday, April 20, 2011

"And nightly Icarus probes his wound/And daily in his workshop, curtains carefully drawn/Constructs small wings and tries to fly/To the lighting fixture on the ceiling: Fails every time and hates himself for trying." -Edward Field, Icarus

Monday, March 28, 2011

What is this feeling? No, it's not loathing. It's not anger, either. For all that I wish one thing or another would happen, I can understand completely why all fell as they did, and I knew they would. This feeling is one of heart-hurt, because knowing doesn't change watching the world's face fall free of the illusion, and also one of sea-shore standing on the banks in the distance, because knowing, knowing does not change framework and expectations, and those glare down at me while I stammer out my explanations that die on leaving my soul.

Saturday, March 26, 2011

To my mind this title is really punny...at least to the first part of this note.

It is 2:13 am. I've decided to pull and all-nighter, except technically it isn't an all-nighter because I think I took a 3 hour nap earlier today. I took a 2 hour nap and a 1 hour nap the two preceeding days and I really don't like the trend that this is going in. Clearly, I am tired - I'm having trouble stringing a coherent and gramatically correct sentence together, and it has taken me nearly 10 minutes to type this. But if I go back to sleep...

1) I'll fall asleep, then sleep and sleep until noon and then I'll lose much of the day and I'll be annoyed and stay up late saturday night and sunday night as well.

2) I'll be tired but I won't be able to fall asleep because there's too much light [there is no light.] or I just won't be able to sleep.

So I'm going to stay awake; however, I need something to do. Most people are asleep. And quite frankly I doubt my ability to hold a conversation with anyone that I know. Recently Borders had this really awesome book clearance, on account of their filing for bankruptcy [which is not so awesome] and I found this nice-looking book on linear algebra. Except nice-looking books on linear algebra aren't what I usually read.

And this is why I think the title is punny.

--

2:38 am

Linear Algebra by Georgi E. Shilov

A Summary [I'll either learn something or wake up with my face plastered to the keyboard with the wisps of a really bizarre dream about numbers.]:

Preface:

"it should be notedthat the term 'linear algebra' has for some time ceased to describe the actualcontent of the course, representing as it does a synthesis of various ideas fromalgebra, geometry and analysis. And although analysis in the strict sense of theterm (i.e., the branch of mathematics concerned with limits, differentiation,integration, etc.) plays only a backgroun role in this book, it is in fact theactual organizing principle of the course, since the problems of "linearalgebra" can be regarded both as "finite-dimensional projections" and as the"support" for the basic problems of analysis."

And it goes on to describe the differences between the author's [Shilov's] previous book [An Introduction to the Theory of Linear Spaces (1961

"LS is entirelyconcerned with real spaces, while this book considers spaces over an arbitrarynumber field,"

and then could vaguely follow

"with the real andcomplex spaces being considered as closely related special cases of the generaltheory,"

before there came a bunch of terms that hopefully will be defined later and maybe I'll find out that they really mean something actually comprehensible but for now I'll just list them: "Jordan canonical form of the matrix of a linear operator in a real or complex space," "canonical form of the matrix of a normal operator in a complex space equipped with a scalar product," "Hermitian, anti-Hermitian and unitary operators [and their real analogues]" and "infinite-dimensional Hilbert space."

Then the author mentions that chapter 11 contains "ancillary material that can be omitted on first reading."

And thanks the editor and an I. Y. Dorfman.

**A definition -- ancillary** Yahoo! Dictionary

an.cil.lar.y (adj) 1. Of secondary importance. 2. Auxiliary.

(n) 1. Something that is subordinate to something else. 2. -Archaic- A servant.

Latin -ancilla- for maidservant, which is a "feminine diminutive of -anculus-" which means servant in "Indo-European roots."

Chapter 1: Determinants

1.1 Number Fields.

1.11 I saw *K* being used as a variable and my eyes glazed over...but I think what the first paragraph is saying is that, like in most of math, linear algebra uses "numbers" (number systems [number fields {any set K of ojects}]). Any set of numbers can be added, subtracted, multiplied, or divided ("subjected to the four arithmetic operations") to make more numbers ("again give elements of K")

Then it lists properties of every pair of numbers A and B in K - commutative, associative, zero, negative element things that take more time to type than it does to memorize -

- for addition (subtraction is a manipulation of addition and negative element)

- for multiplication (division is a manipulation of multiplcation and reciprocal element) *

Natural numbers: 1, 1+1=2, 2+1=3 etc. We assume that none are 0. **

Rational numbers: p/q; p and q are integers and q=/=0

Fields K and K' are isomorphic if

-- Head nod. Must stay awake --

3:01 am

Break time.

Switch to more upbeat radio station. Cut to Doobaba project.

--

3:42 am

Cold. Tired. Not bored, but in the vegetative donotwanttodowork stage.

"Two fields K and K'are said to be isomorphic if we can set up a one-to-one correspondence between Kand K' such that the number associated with every sum (or product) of numbers inK is the sum (or product) of the corresponding numbers in K'. The numberassociated with every difference (or quotient) of numbers in K will then be thedifference (or quotient) of the corresponding numbers in K'."

I have no idea what that means.

Isomorphic=

Two fields. K. K'. [-.-]

K and K' are related.

K and K' = number fields = any set of objects = number systems = group of numbers. Oh.

In each group of numbers... [brain stuttering...failing...fail.]

Sum = +.

Product = x

In K ->

Two numbers: A and B

Sum A and B: A+ B = C

C = number associated with every sum. [I just had an epiphany. {And this is because I am not in a library. (That was probably in bad taste. Sorry.)}]

In K' ->

Two numbers: A' and B'

Sum A' and B': A'+ B' = C'

C = C'

.

...

.

Ugh.

K: A - B = D

K': A' - B' = D'

D = D'

K and K' are isomorphic.

...

I think I'm oversimplifying to the point of wrongness.

--

4:01 am

Search: "isomorphic"

**A definition -- isomorphic** Merriam-Webster

1 a. Being of identical or similar form, shape, or structure.

b. Having sporophytic and gametophytic generations alike in size and shape.

2. related by an isomorphism.

First known use: 1862

** A definition -- isomorphism** ibid.

1: The quality or state of being isomorphic [No. Just no.]

a. Similarity in organisms of different ancestry resulting from convergence.

b. Similarity of crystalline form between chemical compounds.

2: A one-to-one correspondence between two mathematical sets; especially; a homomorphism that is one-to-one - compare ENDOMORPHISM [grrr.]

First known us: circa 1828 [I feel an incongruity here.]

**A definition -- endomorphism** ibid.

A homomorphism that maps a mathematica set into itself - compare ISOMORPHISM.

First known use: 1909

**Wikipedia -- isomorphism**

"In abstract algebra,an isomorphism (Greek: ἴσος isos "equal", and μορφή morphe "shape") is a bijective map f such that both fand its inverse f −1 are homomorphisms, i.e., structure-preserving mappings. In the more general setting of categorytheory, an isomorphism is a morphism f: X → Y in a category for which there exists an "inverse" f −1: Y → X, with the property that both f −1f =idX and f f −1 = idY."

Isomorphism is a morphism. Thank you for informing me. That is probably the only thing I understood.

Informally, anisomorphism is a kind of mapping between objects that shows a relationshipbetween two properties or operations. If there exists an isomorphism betweentwo structures, we call the two structures isomorphic. In a certain sense, isomorphicstructures are structurally identical, if you choose to ignorefiner-grained differences that may arise from how they aredefined.

That explains nothing to me. Right now, I need 1+2=3.

Isomorphisms arestudied in mathematics in order to extend insights from one phenomenon toothers: if two objects are isomorphic, then any property which is preserved byan isomorphism and which is true of one of the objects, is also true of theother. If an isomorphism can be found from a relatively unknown part ofmathematics into some well studied division of mathematics, where many theoremsare already proved, and many methods are already available to find answers, thenthe function can be used to map whole problems out of unfamiliar territory overto "solid ground" where the problem is easier to understand and workwith.

Better.

Consider the group Z6, the integers from 0 to 5 with addition modulo 6. Alsoconsider the group Z2 × Z3, the ordered pairswhere the x coordinates can be 0 or 1, and the y coordinates can be 0,1, or 2, where addition in the x-coordinate is modulo 2 and addition inthe y-coordinate is modulo 3. These structures are isomorphic underaddition, if you identify them using the following scheme:

(0,0) →0

(1,1) →1

(0,2) →2

(1,0) →3

(0,1) →4

(1,2) →5

or in general(a,b) → ( 3a + 4 b ) mod 6. For example notethat (1,1) + (1,0) = (0,1) which translates in the other system as 1 + 3 = 4.Even though these two groups "look" different in that the sets contain differentelements, they are indeed isomorphic: their structures areexactly the same. More generally, the direct product of two cyclic groups Zm and Zn is isomorphic to Zmn if and only if m and n arecoprime.

...I'll just leave it at that.

* Note: in A (B+Y) = AB + AY, for every A, B, Y in K, it is implied that (A+B) Y = Ay + BY.

**

Given two elements Nand E, say, we can construct a field by the rules

N + N=N,

N + E =E,

E + E =N,

N * N =N,

N * E =N,

E * E =E.

Then, in keeping withour notation, we should write N = 0, E = 1, and hence 2 = 1 + 1 = 0. To excludesuch number systems, we should require that all natural field elements benonzero.

End of page 2.

4:40 am.

Very cold.

Also paranoid. Saw large spider near lamp about an hour ago. Forgot about spider. Now spider is not in sight.

About me

I'm me, and that's all I can really be, right?

Well, I don't know. I don't believe in the statement "be true to yourself." Not that I believe in being fickle, or inconsistent. I only mean that I believe in nurture over nature, and that one can change, for better or worse, without knowing it, or even if they so desire.